When I google for complete matching, first link points to perfect matching on wolfram.

It defines perfect matching as follows:

A perfect matching of a graph is a matching (i.e., an independent edge set) in which every vertex of the graph is incident to exactly one edge of the matching. A perfect matching is therefore a matching containing $n/2$ edges (the largest possible), meaning perfect matchings are only possible on graphs with an even number of vertices.

However below graph contains even number of vertices (10 vertices), but still I cant guess the perfect matching - in which every vertex of of the graph is incident to exactly one edge. One of the matching can be $\{E_1,E_4,E_6\}$. However not every vertex (here, $\{V_5,V_6,V_9,V_{10}\}$) is incident to exactly one edge in this matching as required by above definition. So this is not definitely a perfect matching as desired by wolfram's definition. Also I cannot add any other edge to this matching as it will make two edges to share a vertex.

Q1. Is perfect matching possible with this graph?

enter image description here

I am also reading the same topic from the book by Narsingh Deo. It defines the complete matching in the context of bipartite graph:

In a bipartite graph having a vertex partition $V_1$ and $V_2$ a complete matching of vertices in set $V_1$ into those in $V_2$ is a matching in which there is one edge incident with every vertex in $V_1$. In other words, every vertex in $V_1$ is matched against some vertex in $V_2$.

I am not able to understand if these two (wolfram's and book's) definitions point to two different concepts or they are one and same.

By my understanding, according to Deo's definition, $V_2$ may contain more vertices than that in $V_1$, thus the size matching may be less than $n/2$. However, wolfram says perfect matching contains $n/2$ edges. For example in above graph the complete matching of vertices in set $P_1$ into those in $P_2$ can be $\{E_1,E_4,E_6\}$ as one edge in this matching is incident with every vertex in $P_1$.

Q2. So the two definitions (wolfram's and book's) must be different concepts. Am I correct?


These are two different concepts. A perfect matching is a matching involving all the vertices. A bipartite perfect matching (especially in the context of Hall's theorem) is a matching in a bipartite graph which involves completely one of the bipartitions. If the bipartite graph is balanced – both bipartitions have the same number of vertices – then the concepts coincide.

The term bipartite perfect matching is not standard – in fact I just made it up. Usually perfect matching just means a matching involving all the vertices.

  • $\begingroup$ +1 for the term "bipartite perfect matching", serves the purpose of distinguishing the two concepts and also gives identity to the this particular concept, though is simple one $\endgroup$ – Maha Dec 7 '15 at 7:43
  • $\begingroup$ Just as an addendum to a sufficient answer, the comparitively more frequent term for the aforementioned 'bipartite perfect matching' is 'X-saturating matching'. $\endgroup$ – Shiven Sinha Apr 26 at 15:08

Yuval is right. A perfect matching in a bipartite graph, may be restricted and defined differently as a matching, which covers only one part of the graph. Note that according to such a definition, the number of vertices in the graph may be odd. For example, you can delete say $V_{10}$ of your example, and your matching is still perfect in the restricted sense.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.